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 A059070 Card-matching numbers (Dinner-Diner matching numbers). 0
 1, 44, 45, 20, 10, 0, 1, 13756, 30480, 32365, 21760, 10300, 3568, 970, 160, 40, 0, 1, 6699824, 23123880, 38358540, 40563765, 30573900, 17399178, 7723640, 2729295, 776520, 180100, 33372, 5355, 540, 90, 0, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This is a triangle of card matching numbers. A deck has 5 kinds of cards, n of each kind. The deck is shuffled and dealt in to 5 hands with each with n cards. A match occurs for every card in the j-th hand of kind j. Triangle T(n,k) is the number of ways of achieving exactly k matches (k=0..5n). The probability of exactly k matches is T(n,k)/((5n)!/n!^5). Rows are of length 1,6,11,16,... REFERENCES F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12. S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617-620. J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178. R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71. LINKS F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197. B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118. Barbara H. Margolius, Dinner-Diner Matching Probabilities FORMULA G.f.: sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k) where n is the number of kinds of cards (5 in this case), k is the number of cards of each kind and R(x, n, k) is the rook polynomial given by R(x, n, k)=(k!^2*sum(x^j/((k-j)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the coefficient for x^j of the rook polynomial. EXAMPLE There are 32365 ways of matching exactly 2 cards when there are 2 cards of each kind and 5 kinds of card so T(2,2)=32365. MAPLE p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k); for n from 0 to 4 do seq(coeff(f(t, 5, n), t, m)/n!^5, m=0..5*n); od; MATHEMATICA p[x_, k_] := k!^2*Sum[x^j/((k - j)!^2*j!), {j, 0, k}]; r[x_, n_, k_] := p[x, k]^n; f[t_, n_, k_] := Sum[Coefficient[r[x, n, k], x, j]*(t - 1)^j*(n*k - j)!, {j, 0, n*k}]; Table[ Coefficient[f[t, 5, n], t, m]/n!^5, {n, 0, 4}, {m, 0, 5*n}] // Flatten (* Jean-François Alcover, Oct 21 2013, after Maple *) CROSSREFS Cf. A008290, A059056-A059071. Sequence in context: A165865 A239534 A171906 * A059071 A085519 A112815 Adjacent sequences:  A059067 A059068 A059069 * A059071 A059072 A059073 KEYWORD nonn,tabf,nice AUTHOR Barbara Haas Margolius (margolius(AT)math.csuohio.edu) STATUS approved

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Last modified December 6 21:48 EST 2019. Contains 329809 sequences. (Running on oeis4.)